Let’s say we have representations and . That is, we have functors and with , — where is the single object of , when it’s considered as a category — and the given actions on morphisms. We want to consider a natural transformation .

Such a natural transformation consists of a list of morphisms indexed by the objects of the category . But has only one object: . Thus we only have one morphism, , which we will just call .

Now we must impose the naturality condition. For each arrow in we ask that the diagram

commute. That is, we want for every algebra element . We call such a transformation an “intertwiner” of the representations. These intertwiners are the morphisms in the category of of representations of . If we want to be more particular about the base field, we might also write .

Here’s another way of putting it. Think of as a “translation” from to . If is an isomorphism of vector spaces, for instance, it could be a change of basis. We want to take a transformation from the algebra and apply it, and we also want to translate. We could first apply the transformation in , using the representation , and then translate to . Or we could first translate from to and then apply the transformation, now using the representation . Our condition is that either order gives the same result, no matter which element of we’re considering.

[…] of . Then since the symmetrizer and antisymmetrizer are elements of the group algebra , they define intertwiners from to itself. The their images are not just subspaces on which the symmetric group acts nicely, […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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